Nothing in real time graphics _actually_ is cross platform. You are always programming a vendor hardware.
"Cross platform" really has no value in graphics programming, honestly, when discussing the lowest level programmable api.
You always need to explicitly say which platforms you target, and then test on those platforms.
The main questions should be "on which platform will my users run it" and "can I debug and test this".
So if you are starting out - pick out the easiest api for you. Don't go looking for universality as a value on it's owm (if you want universal rendering, choose CPU rendering. if you want high perf - choose your GPU platform(s) explicitly).
The "universal" nature of WebGPU with the "web" focus feels a lot like the graphics API equivalent of WASM. In that regard there are also some interesting projects/opportunities.
Having shipped non-trivial WASM code in two projects That's not a positive thing. WASM is stuck in zone between living and the dead - just good enough (barely) to merit creating production code when you need the perf in browser but not good enough to register as industrial quality tool as it's not possible really to debug a WASM application.
If you really, really need the perf in your browser app, WASM may be the only option, but it's not really a good one.